A Shell Element Based on SE(3) Group with Precision-Reserved Interpolation for Geometrically Nonlinear Problems

Based on the special Euclidean group SE(3), a geometrically exact shell element is proposed for the analysis of structures undergoing large deformation and finite rotation. First, a unified description of the nodal variables is established within the SE(3) framework, which accurately captures the coupling effect of translation and rotation. However, conventional interpolation schemes on the non-commutative manifold are path-dependent and fail to maintain physical objectivity, which often leads to spurious strain energy. By combining implicit iterative interpolation and explicit relative configuration interpolation, a precision-reserved interpolation scheme is proposed. By applying the logarithmic mapping on the SE(3) manifold, nodal configuration increments are transformed into the left tangent space of the same reference point, eliminating path dependency. Subsequently, the Lagrange interpolation is applied to both the translational and rotational increments in this tangent space, ensuring their C° continuity. Finally, the explicit expressions for the discrete deformation gradients and strains are derived based on the variational principles. Furthermore, the permutation tensor is utilized to handle the variation and linearization of the involved nonlinear mappings. It results in the explicit expression for the geometric stiffness matrix and thus reduces the updating operation of the Jacobian matrix during iterations. Four numerical examples are presented to verify the property of the element in resisting shear locking and its accuracy in handling geometric nonlinear problems of thin-walled or thick-walled structures.